A couple passages from JM's memoir suggests it's more him writing independently, with Morrissey coming in later:
New songs usually started with me recording the music on to a cassette and then giving it to Morrissey to write his lyrics and vocal lines, which he would complete within a day or two. Other times we would get together at my place and sit face to face, about three feet away from each other, while I played my new tune into a tape recorder that was balanced between my knees.
As I went around and around, the tune started to get psychedelic in my headphones and I knew I was on to something. I programmed a simple beat on the “drum machine and recorded the hypnotic rhythm guitar, and then came up with a two-note phrase that I put on top. What I’d done was nothing like the other two songs, and nothing like anything the band had done before either. When Angie got back I played her the demo and she thought it was great. Then I took the cassette of the three songs round to Morrissey’s, having written ‘Fast’, ‘Irish Waltz’ and ‘Swampy’ on it. He worked on it for a few days, and when he’d finished the lyrics the songs became ‘William, It Was Really Nothing’, ‘Please, Please, Please, Let Me Get What I Want’ and ‘How Soon Is Now?’.
He does say, however, that with some of their earliest songs he received words from Morrissey and worked from there.
"Ain't no party like an S Club party!'" - Richard Nixon, Checkers Speech, abandoned early draft